Complex eigenvalue instantons and the Fredholm determinant expansion in the Gross-Witten-Wadia model

Abstract

We study the leading nonperturbative corrections to the strong-coupling (ungapped) phase of the Gross-Witten-Wadia (GWW) integral over unitary matrices, to one-loop order. We compute these corrections directly in terms of eigenvalue tunneling in a holomorphic presentation of the integral over eigenvalues. The leading nonperturbative contribution to the partition function comes from a pair of complex eigenvalue instantons. We show that these are in fact “ghost instantons”, which are extrema of the one-eigenvalue effective potential on the “unphysical sheet” of the spectral curve and have been discussed in detail recently by Mariño, Schiappa, and Schwick. Further, we discuss the relationship of these instantons to the Fredholm determinant expansion of the unitary matrix integral, which has recently become an object of interest in the computations of BPS indices of supersymmetric gauge theories and black holes. We find that, after taking the ’t Hooft limit, the first correction given by the Fredholm determinant expansion of the GWW integral agrees precisely with the leading nonperturbative correction, to one-loop order.

Normalization of ZZ instanton amplitudes in minimal string theory

“…These matrix integrals also exhibit two phases, separated by a gap-closing phase transition in the eigenvalue density, the simplest example of which, at k " 1, is the double-scaled version of the Gross-Witten-Wadia transition [33,34,35]. We generalize the matrix computation of non-perturbative corrections to the partition function from the k " 1 case [36,37,38] to general k in both phases, and find perfect agreement with the dual string theory results.…”

“…In contrast with the situation in the gapped phase, where the leading instanton contributions to the partition function consisted of single-eigenvalue tunneling, in the ungapped phase, the leading corrections are due to pairs of instantons [38] consisting of a positive-charge and a negative-charge instanton. We will defer the details of how to obtain these contributions to appendix B, and just note that the one-loop normalization factor associated to a pair of instantons located at ξ 1 and ξ 2 is…”

“…This is a well-known fact in the Gross-Witten-Wadia model [36]. From the matrix integral perspective, this is because the one-loop prefactor multiplying the one-instanton contribution vanishes [38]. It is the purpose of this subsection to argue this independently on the string theory side.…”

“…The physical intuition is that summing over an infinite number of closed string states can capture the change in the RR flux. The analog of this in the matrix integral is that if we compute an observable like det U in the ungapped phase, it does have a one-instanton correction [37,38].…”

Giant graviton expansion of Schur index and quasimodular forms

“…We follow the conventions in[23] where, in particular, details on the BPS condition can be found 3. It is possible to discuss (1.2) in the U(N ) SYM theory without reference to its string dual description[24][25][26],up to possible non-trivial reorderings of the sum[27].…”